In this section, we would like to study the HMI for MAdS background (). Since the HMI is a linear combination of HEE we consider the special limit that the HEE can be computed analytically. Considering the high energy limit i.e. \(l\Lambda _c \ \& \ l^{\prime }\Lambda _c \ll 1\)  or equivalently \(r_c\ll r^*_l \ \& \ r^*_{l^{\prime }}\)  and using eqs. (REF ) and (), we get

I(lc ,l'c )4GN(5) I(l,l',lc,l'c)L2c2=1[2(lc)2-1(l'c)2-1(l'c+2lc)2]-23(l'c+lc)2,      1<0,3>0,

where \(\hat{\mathcal {I}}(l\Lambda _c ,l^{\prime }\Lambda _c )\)  is the redefined \(I(l,l^{\prime },l\Lambda _c,l^{\prime }\Lambda _c)\)  which is given by eq.(REF ) in appendix . By this redefinition, the limits of \(l\Lambda \rightarrow 0\)  and \(l^{\prime }\Lambda _C\rightarrow 0\)  become meaningful. The first term in eq.(REF ), including \(\kappa _1\) , is the contribution of the AdS boundary corresponds to the mutual information between two subsystems in conformal field theory [1]} and the second term, including \(\kappa _3\) , is the non-conformal effect. Since \(\kappa _3>0\)  is a positive constant the second term in eq.(REF ) are always negative and hence the non-conformality decreases \(\hat{\mathcal {I}}(l\Lambda _c ,l^{\prime }\Lambda _c )\) . From the above equation, we observe that by increasing \(l\Lambda _c\)  (\(l^{\prime }\Lambda _c\) ) \(\hat{\mathcal {I}}(l\Lambda _c ,l^{\prime }\Lambda _c )\)  increase (decrease) for fixed \(l^{\prime }\Lambda _c\)  (\(l\Lambda _c\) ) in the high energy limit.
